 A cylinder makes six turns in 2 seconds, calculate: a) its angular velocity in rad/s; b) its period and c) its frequency.

a) Angular velocity (\omega) is given by the formula:

\omega = \frac{\Delta \theta}{\Delta t}

Here, the cylinder makes 6 turns, and each turn is 2\pi radians. Therefore, in 6 turns, the angle in radians is:

\Delta \theta = 6 \times 2\pi = 12\pi \text{ radians}

The time period (\Delta t) is 2 seconds, so the angular velocity is:

\omega = \frac{12\pi}{2} = 6\pi \text{ rad/s}

b) The period (T) is the time it takes to complete one full rotation (1 turn). Since the cylinder makes 6 turns in 2 seconds:

T = \frac{\Delta t}{\text{number of turns}} = \frac{2}{6} = \frac{1}{3} \text{ s}

c) The frequency (f) is the reciprocal of the period:

f = \frac{1}{T} = \frac{1}{\frac{1}{3}} = 3 \text{ Hz}

Thus, the answers are:

a) \omega = 6\pi \text{ rad/s}

b) T = \frac{1}{3} \text{ s}

c) f = 3 \text{ Hz}